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Consider the limited attention choice framework by Matejka and McKay (2015).

This framework can give rise to consideration sets, as roughly summarised below.

Consideration sets in the limited attention choice framework: Since attention is a scarce resource, the agents may be induced to contemplate only a subset of the available alternatives, ignoring all the others. More precisely, this framework assumes that the decision maker (hereafter, DM) knows the feasible set but is not fully aware of the value of each alternative in the feasible set. The DM is endowed with a prior on the value of each alternative in the feasible set and can "buy" an information structure to update such a prior. An information structure consists of a signal that, once realised, allows the DM to update her prior and then choose the alternative from the feasible set maximising the expected utility given the refined information. On one hand, the DM prefers a signal that is as informative as possible on the value of each alternative in order to minimise ex-post regret. On the other hand, limited attention constraints imply that a more informative signal is more costly because it requires more cognitive effort to be processed. Such a trade-off can lead to equilibria featuring information structures such that some alternatives in the feasible set are never chosen by the DM for any realisation of the signal. Hence, these alternatives are not part of the DM's consideration set. Complementarily, the DM consideration set's consists of the alternatives which are chosen by the DM with positive probability for at least some realisations of the signal.

Question: Consider the Sen's property $\alpha$ stating that if an alternative $x$ is chosen from a set $T$, and $x$ is also an element of a subset $S$ of $T$, then $x$ must be chosen from $S$. Does this property hold in the the limited attention choice framework by Matejka and McKay (2015)? The authors have a discussion about IIA (independence of irrelevant alternatives) but I couldn't understand whether the Sen's property $\alpha$ holds or not.

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My math is a bit rusty, but I believe you can show that, for a well-behaved (e.g., open and connected) feasible set $T$ and a consideration set $T'$, $x \in S \subseteq T' \subseteq T$, e.g. Sen's property $\alpha$ holds.

That doesn't seem like a very interesting question/result though, as it just follows directly from the definition of "well-behaved". Maybe I'm missing something.

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